ABSTRACT. The aim of this paper is to establish similar results due to G. Bennett[2] and C. P. Niculescu[4] for newly defined class.

Transcription

1 On the G. ennett s inequality Imran bbas aloch bdus Salam School of Mathematical Sciences G University, Lahe, akistan. iabbasbaloch@gmail.com [2010]rimary: 26D15. Secondary: 2651 onvex function, el measure, barycenter, 3-convex functions at a point. STT The aim of this paper is to establish similar results due to G. ennett[2].. Niculescu[4] f newly defined class. 1. INTODUTION vast they evolved in the study of convex functions, may be readily appealed to most significant topics in real analysis economics. In past recent years, a speedy development has encontered in the they of convex functions. This can be endsed to several causes, two of which are as follows: First, so many sections in modern analysis directly indirectly covers application of convex functions. Second, convex functions have huge impact on the they of inequalities several imptant inequalities are consequences of the application of convex functions (see [5]). In [2], G. ennett presented some out-turns of an inequality which illustrate about the behavi of convex functions with respect to a mass distribution. Later on,.. Niculescu established an abstract version of this inequality([4]), which is shown in the following theem Theem 1. [4] Let I is an interval carrying a +ve el measure l ; ; are three disjoint compact subintervals of I of +ve measure. Then (1.1) l() = l( ) + l( ) (1.2) αdl(α) = αdl(α) + αdl(α); give a necessary sufficient condition under which the inequality (1.3) f (α)dl(α) f (α)dl(α) + is valid f every convex function f : I. f (α)dl(α). Definition 1. ([4] also see [4], p. 179, f details). ny real el measure l on an interval I such that l(i ) > 0 f (α)dl(α) 0 f every nonnegative conveα f unction f : I. is called a Steffensen-opoviciu measure. I brief description of this concept is given in the following result, independently due to T. opoviciu [6]. M. Fink [3]: Lemma 1. Suppose that l be a real el measure on an interval I with l(i ) > 0. Then l is a Steffensen- opoviciu measure iff the following condition of endpoints positivity, (t α)dl(α) 0 (α t)dl(α) 0 holds f every t [a, b]. I (,t] 1 I [t, )

2 2 Theem 2. ([4] also see [4], p , f details) Let l is a Steffensen- opoviciu measure on an interval I. Then the inequality f (b l ) 1 l(i ) I f (α)dl(α) holds f every continuous convex function f on I, here b l = 1 l(i ) I αdl(α) represents the barycenter of l. Definition 2. [4] ny real el measure l on an interval I such that l(i ) > 0 f (α)dl(α) 0 f every nonnegative concave f unction f : I. I is called a dual Steffensen-opoviciu measure. Theem 3. Theem 1 even wks if l is a real el measure on I,, are three disjoint subintervals of I such that the restriction of l to each of the intervals is a Steffensen-opoviciu measure the restriction of l to is a dual Steffensen-opoviciu measure. Now, we consider the inequality of G. ennett f new class of functions in following section. 2. MIN ESULTS In [1], I.. aloch, J. ečarič, M. raljak defined a new class of functions which is defined as follow Definition 3. Let c I, where I is an open, closed semi-open interval in either direction in I is its interi. We say that f : I is 3-convex function in point c (respectively 3-concave function at point c) if there exists a constant K such that the function F(α) = f (α) K 2 α2 is concave (resp. convex) on I (,c] convex (resp. concave) on I [c, ). f is 3-concave function in point c if f is 3-convex function in point c. property that explains the name of the class is the fact that a function is 3-convex on an interval if only if it is 3-convex at every point of the interval (see [1]). Theem 4. Let I is an interval bearing a +ve el measure l ; ; ; ; ; are six disjoint compact subintervals of I of +ve measure such that (2.1) l() = l( ) + l( ) ; l() = l() + l() (2.2) αdl(α) = αdl(α) + αdl(α) ; also c I is such that βdl(β) = βdl(β) + βdl(β). (2.3) max{right end points o f interval,, } c min{le ft end points o f interval,,} Now, if (2.4) then following inequality (2.5) f (α)dl(α) + α 2 dl(t) + α 2 dl(α) α 2 dl(α) = β 2 dl(β) + β 2 dl(β) β 2 dl(β), holds f every 3-convex function f : I at a point c. f (α)dl(α) f (α)dl(α) f (β)dl(β) + f (β)dl(β) f (β)dl(β) roof. Since f is 3-convex function at point c I, then we have a constant K such that F(α) = f (α) K 2 α2 is concave on I (,c]. Therefe, f ; ;, three disjoint compact subintervals of I [c, ) of +ve measure, so by reverse of the inequality (1.3), we have 0 F(α)dl(α) + F(α)dl(α) F(α)dl(α) = f (α)dl(α) + f (α)dl(α) f (α)dl(α) K ( 2

3 3 lso, by using the fact that F(β) = f (β) K 2 β 2 is convex on I [c, ). Therefe, f ; ;, three disjoint compact subintervals of I [c, ) of +ve measure, so by use of the inequality (1.3), we have 0 F(β)dl(β) + F(β)dl(β) F(β)dl(β) = f (β)dl(β) + f (β)dl(β) f (β)dl(β) K ( 2 From above, we have f (α)dl(α) + f (α)dl(α) f (α)dl(α) K ( 2 0 f (β)dl(β) + f (β)dl(β) f (β)dl(β) K ( 2 So, f (α)dl(α) + f (α)dl(α) f (α)dl(α) K ( 2 (2.6) f (β)dl(β) + f (β)dl(β) f (β)dl(β) K ( 2 y using (2.4), we get (2.5). emark 1. From the proof of the Theem 4, we have (2.7) f (α)dl(α) + f (α)dl(α) f (α)dl(α) K ( 2 (2.8) f (β)dl(β) + f (β)dl(β) f (β)dl(β) K ( 2 So under assumption (2.4), we can get a improvement of (2.5) as follow f (α)dl(α) + f (α)dl(α) f (α)dl(α) K ( 2 (2.9) { = K 2 ( β 2 dl(β) + β 2 dl(β) β 2 dl(β) )} f (β)dl(β) + f (β)dl(β) f (β)dl(β) Now, we give next result which weakens the assumption (2.4) such that inequality (2.5) holds under this new condition. Theem 5. Suppose that I is an interval bearing a +ve el measure l ; ; ; ; ; are six disjoint compact subintervals of I of +ve measure such that (2.1) (2.2) hold with (2.10) a = max{right end points o f interval,, } min{le ft end points o f interval,,} = b f is 3-convex at a point c f some c [a,b]. Then if (a) f (a) 0 (b) α 2 dl(α) + α 2 dl(α) α 2 dl(α) β 2 dl(β) + β 2 dl(β) β 2 dl(β) f +(b) 0 α 2 dl(α) + α 2 dl(α) α 2 dl(α) β 2 dl(β) + β 2 dl(β) β 2 dl(β)

4 4 (c) then (2.5) holds. (a) < 0 < +(b) f is 3 convex, roof. The idea of proof is similar to proof of Theem 4. Hence, by proceeding as in the proof of Theem 4. From the inequality 2.6, we have K [( ( ] 2 f (β)dl(β) + f (β)dl(β) f (β)dl(β) ( f (α)dl(α) + f (α)dl(α) f (α)dl(α) ) Now, due to the concavity of F on I (,c] convexity of F on I [c, ), so f every distinct points α j I (,a] y j I [b, ), j = 1,2,3, we have Letting α j a β j b, we get (if exists) [α 1,α 2,α 3 ] f K [β 1,β 2,β 3 ] f f (a) K f +(b) Therefe, if assumptions (a) (b) holds, then K [( ( ] 2 is positive we conclude the result. If the assumption (c) holds, the f is left continuous, f + is right continuous, they are both non-decreasing f f +. Therefe, there exists c [a,b] such that f with associated constant K = 0 again, we can deduce the result. emark 2. gain from the proof of Theem 5, we obtain the inequalities (2.7) (2.8). Now, under assumption (a), (b) (c) of Theem 5, K is positive negative zero respectively due to argument discussed in the proof. Therefe, we get a better improvement of (2.5) then (2.9) in this case as follow f (α)dl(α) + f (α)dl(α) f (α)dl(α) K ( 2 (2.11) K 2 ( β 2 dl(β) + β 2 dl(β) β 2 dl(β) ) f (β)dl(β) + f (β)dl(β) f (β)dl(β) Under the assumption of Theem 4 with f : I is 3-concave at point c I, the reverse of inequality (2.5) holds. Now, we give only the statement of the theem with weaker condition which can be proved in similar way under which the reverse of inequality (2.5) holds f f : I is 3-concave at point c I. Theem 6. Suppose that I is an interval bearing a +ve el measure l ; ; ; ; ; are six disjoint compact subintervals of I of +ve measure such that (2.1) (2.2) hold with (2.12) a = max{right end points o f interval,, } min{le ft end points o f interval,,} = b f : I is 3-concave at a point c f some c [a,b]. Then if (a) f (a) 0 (b) α 2 dl(α) + α 2 dl(α) α 2 dl(α) β 2 dl(β) + β 2 dl(β) β 2 dl(β) +(b) 0 α 2 dl(α) + α 2 dl(α) α 2 dl(α) β 2 dl(β) + β 2 dl(β) β 2 dl(β)

5 5 (c) then reverse of (2.5) holds. (a) < 0 < +(b) f is 3 concave, emark 3. Similarly as in emark 2, we obtain the reverse of inequalities (2.7) (2.8) from the proof of Theem 6. Now, due the convexity of F on I (,c] concavity of F on I [c, ), so f every distinct points α j I (,a] β j I [b, ), j = 1,2,3., we have Letting α j a β j b, we get (if exists) [ α 1, α 2, α 3 ] f K [ β 1, β 2, β 3 ] f f (a) K f +(b) Now, under assumption (a) (b) (c) of Theem 5, K is negative positive zero respectively due to argument discussed above. Therefe, we get a better improvement in this case as follow f (α)dl(α) + f (α)dl(α) f (α)dl(α) K ( 2 (2.13) K ( f (β)dl(β) + f (β)dl(β) f (β)dl(β) 2 Theem 7. Suppose that I is an interval bearing a el measure l ; ; ; ; ; are six disjoint subintervals of I with the restriction of l to each of the intervals,, is a Steffensen-opoviciu measure the restriction of l to is a dual Steffensen-opoviciu measure such that (2.14) l() = l( ) + l( ) ; l() = l() + l() (2.15) αdl(α) = αdl(α) + αdl(α) ; also c I is such that βdl(β) = βdl(β) + βdl(β). (2.16) max{right end points o f interval,,i } c min{le ft end points o f interval,,} Now, if (2.17) then following inequality (2.18) f (α)dl(α) + α 2 dl(α) + α 2 dl(α) α 2 dl(α) = β 2 dl(β) + β 2 dl(β) β 2 dl(β), holds f every f : I 3-convex function at a point c. f (α)dl(α) f (α)dl(α) f (β)dl(β) + f (β)dl(β) f (β)dl(β) EFEENES [1] I.. aloch, J. ečarič, M. raljak,generalization of Levinson s inequality. J. Math. Inequal. 9(2), (2015). [2] G. ennett, p-free l p Inequalities,mer. Math. Monthly 117(2010), No.4, [3]. M. Fink, best possible Hadamard inequality, Math. Inequal. ppl. 1(1998), [4].. Niculescu, On result of G. ennett, ull. Math. Soc. Sci.Math. oumanie Tome54(102), No.3, 2011, [5] J. ečarič,,f. roschan, Y.L. Tong : onvex functions, artial derings Statistical application. cadmec ress, New Yk(1992). [6] T. opoviciu, Notes sur les fonctions convexes d dre superieur (IX), ull. Math. Soc.oum. Sci. 43(1941),